This extended persistence can be broken down into multiple components [1]}, the ordinary part, formed of topological features that are both born and die within the homology groups of the sublevel sets of \(X\) , the relative part of features that are born and die in the relative homology groups, and the extended part of features that are born in the ordinary homology groups and die in the relative homology groups in the filtration. The birth time \(b\)  of a feature may be larger than its death time \(d\)  due to the possibility that the feature dies in the relative homology group \(H(X,X^d)\)  with \(d<b\) . The extended part can be further divided into topological features that have \(b<d\) , termed extended\(+\) , and those with \(d<b\) , termed extended\(-\) .
